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The ROCs associated with the indices η, Q, and LOR, and with equal- variance logistic PDFs in detection theory, are demonstrated here to be hyperU+00A- bolas, according to a proof by C. E. Metz (personal communication, 1984).
The common functional form of these ROCs (as shown in Table 3) iswhere c is a nonnegative constant (< 1) specifying the ROC+ADs- a smaller c implies a higher ROC. These ROCs have maximum slope, 1/c, at (0, 0) and minimum slope, c, at (1, 1).
If the coordinates are shifted from the x, y system to a u, v system having the center of the hyperbola at its origin, in particular, such thatso that the origin of the u, v system is to the left of and above the (0, 1) point in the x, y system, then